Schrödinger equation

Question

How to prove that map $f\mapsto u$ from initial value to solution of Schrödinger equation is continuous map of $S(R^n)$ to $C^\infty(R^n,R)$? Thanks in advance.

Answer 1

$$ \hat u(x,t) = \int_{\mathbb R^n} \exp(it|x-y|^2) \hat f(y) \, dy \\= \exp(i|x|^2) \int_{\mathbb R^n} \exp(-2it x\cdot y) \exp(it|y|^2) \hat u(y) \, dy \\= \exp(i|x|^2) \hat h(2tx) ,$$ where $h(x) = \exp(it|x|^2) \hat u(x)$. Now use the fact that the Fourier transform, and multiplying by $\exp(it|x|^2)$, both map $\mathcal S$ into $\mathcal S$.